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On some problems of K. Borsuk concerning shape dominations. (English) Zbl 0834.55009

Nuriev, B. R. (ed.), Baku international topological conference held at Baku (USSR), October 3-9, 1987. Proceedings. Baku: Ehlm, 243-246 (1989).
The capacity \(C(A)\) of a compactum \(A\) is defined as the cardinality of the class consisting of shapes of all compacta \(X\) for which \(Sh(X) \leq Sh(A)\). A system consisting of \(k\) compacta \(A_1, A_2, \dots, A_k\) is said to be a chain of length \(k\) for a compactum \(A\) if \(Sh(A_1) < Sh(A_2) < \cdots < Sh(A_k) \leq Sh(A)\). The depth \(D(A)\) of a compactum \(A\) is defined as the upper bound of lengths of all chains for \(A\). The author gives negative answers to the following questions of K. Borsuk [Russ. Math. Surv. 34, No. 6, 24-26 (1979); translation from Usp. Mat. Nauk 34, No. 6(210), 23-25 (1979; Zbl 0436.55009)]:
1. Is \(C(A \cup B)\) determined by \(C(A)\), \(C(B)\) and \(C(A \cap B)\)?
2. Is \(D(A \cup B)\) determined by \(D(A)\), \(D(B)\) and \(D(A \cap B)\)?
3. Is \(C(A \times B)\) determined by \(C(A)\) and \(C(B)\)?
4. Is \(D(A \times B)\) determined by \(D(A)\) and \(C(B)\)?
5. Is \(C(A)\) determined by the homology properties of \(A\)?
6. Is it true that the capacity of every plane compactum is \(\leq \aleph_0\)?
For the entire collection see [Zbl 0742.00083].

MSC:

55P55 Shape theory
54C56 Shape theory in general topology
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